APPLIED TIME SERIES ANALYSIS

The importance of temporal data analysis is widely recognized, be it observational data, outputs of climate models or paleoclimatic reconstructions.
The course aims to give the elements to autonomously conduct a data analysis, both by providing theoretical insights and operational tools (software).
The course starts with a brief revision of the main concepts in probability and statistics. We continue with the descriptive analysis of time series. A part of the course will be dedicated to modeling time series using the flexible class of ARMA models. More advanced topics that will be covered in the following are spectral analysis, time series regression and multivariate analysis.
The course is part of a three-course statistics path that is offered to doctoral students of the Department: the path is composed of a first introductory course PHD154 Data Analysis with Statistical Programming which precedes two more advanced non-sequential courses (PHD076 Statistical Methods for Climate Change Analysis and this course). Students who are interested in gaining a more solid background in statistical sciences are highly encouraged to follow all courses and to discuss with the instructor of the course for their Ph.D. program the best course of action.

After completing the course successfully, the student should:

- Be able to recognize time-dependent data and describe its important features.
- Have the prerequisite background to define, explain and use terminology such as trend, seasonality, correlated errors and periodicity.
- Be able to apply the commonly used statistical and computational time series techniques to analyze data and make inferences such as estimation and forecasts.
- Understand the statistical methodology underlying the data analysis of time series data, the most important time series models and their properties.

No formal pre-requisite. The course will make use of some mathematics and statistics concepts such as functions, integrals, derivatives, matrices, distributions, estimation and hypothesis testing. Thus students should have the background of an elementary statistics course (at STAT-100 level).
During the course we will use the R software and an elementary knowledge of this software would be useful. Students who feel they do not have the necessary familiarity with these topics are highly encouraged to follow the course PHD154 Data Analysis with Statistical Programming which is offered in the Environmental Sciences Ph.D. program and to discuss with the course instructor the best course of action for their learning.

1. Distribution functions and probability density functions
2. Statistical estimation of parameters
3. Significance tests.
4. Introduction to R (part I)
5. Introduction to R (Part II)
6. Linear regression part I
7. Linear regression part II
8. Stationary stochastic processes and ergodicity
9. ARMA and ARIMA processes
10. Spectral representation
11. Methods of spectral estimation
12. Time Series Regression
13. Multivariate analysis
14. Principal Component Analysis and empirical orthogonal functions (EOF)
15. Alternative analyzing techniques ( Canonical Correlation Analysis, Singular Value Decomposition)

Main text

Shumway, Robert H., Stoffer, David S. (2017) Time Series Analysis and Its Applications With R Examples, Springer (the authors' version of the book can be found in https://www.stat.pitt.edu/stoffer/tsa4/tsa4.htm )

Other useful references

Pruscha, E. (2013), Statistical Analysis of Climate Series, Springer.
Chandler, R.E., Scott, M.E. (2011) Statistical Methods for Trend Detection and Analysis in the Environmental Sciences, Wiley.

Project: Each student will find a time series data set of his/her choice and analyze, model the series using methods learned in this course. The analysis will be written up in a final report with abstract, body text, conclusions, and appendices.
Further information about the project will be disseminated later. The project will be discussed during an oral presentation

The course consists of a combination of conventional theoretical classes focused on a description of methods and practice sessions describing the implementation and application of the methods to real problems. Methods will be implemented with the statistical language R (www.r-project.org). Students are encouraged to bring their own laptops and to experience with the code during some parts of the lessons.
Homework: there will be periodic homework assignments during the course.
Personal participation is important, and it is will help the student to learn more efficiently to read the assigned material to reinforce the lectures.

English

No

oral